If I'm reading these cards correctly, the Universe Starter Spider-man with both Pounce and Unstoppable would be able to ignore hindering terrain while making his pounce. If this right, would he still be able to breakdown blocking terrain at the same time?

Correct, he may ignore hindering terrain while Pouncing if he also has Unstoppable.

However, you would not get the free action to attack blocking terrain, as you only recieve that during a Move Action, and Pounce is a Power Action.